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Simplifying 0.198x2 + 0.054x + -0.461 = 0 Reorder the terms: -0.461 + 0.054x + 0.198x2 = 0 Solving -0.461 + 0.054x + 0.198x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.198 the coefficient of the squared term: Divide each side by '0.198'. -2.328282828 + 0.2727272727x + x2 = 0 Move the constant term to the right: Add '2.328282828' to each side of the equation. -2.328282828 + 0.2727272727x + 2.328282828 + x2 = 0 + 2.328282828 Reorder the terms: -2.328282828 + 2.328282828 + 0.2727272727x + x2 = 0 + 2.328282828 Combine like terms: -2.328282828 + 2.328282828 = 0.000000000 0.000000000 + 0.2727272727x + x2 = 0 + 2.328282828 0.2727272727x + x2 = 0 + 2.328282828 Combine like terms: 0 + 2.328282828 = 2.328282828 0.2727272727x + x2 = 2.328282828 The x term is 0.2727272727x. Take half its coefficient (0.1363636364). Square it (0.01859504133) and add it to both sides. Add '0.01859504133' to each side of the equation. 0.2727272727x + 0.01859504133 + x2 = 2.328282828 + 0.01859504133 Reorder the terms: 0.01859504133 + 0.2727272727x + x2 = 2.328282828 + 0.01859504133 Combine like terms: 2.328282828 + 0.01859504133 = 2.34687786933 0.01859504133 + 0.2727272727x + x2 = 2.34687786933 Factor a perfect square on the left side: (x + 0.1363636364)(x + 0.1363636364) = 2.34687786933 Calculate the square root of the right side: 1.531952306 Break this problem into two subproblems by setting (x + 0.1363636364) equal to 1.531952306 and -1.531952306.Subproblem 1
x + 0.1363636364 = 1.531952306 Simplifying x + 0.1363636364 = 1.531952306 Reorder the terms: 0.1363636364 + x = 1.531952306 Solving 0.1363636364 + x = 1.531952306 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + x = 1.531952306 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + x = 1.531952306 + -0.1363636364 x = 1.531952306 + -0.1363636364 Combine like terms: 1.531952306 + -0.1363636364 = 1.3955886696 x = 1.3955886696 Simplifying x = 1.3955886696Subproblem 2
x + 0.1363636364 = -1.531952306 Simplifying x + 0.1363636364 = -1.531952306 Reorder the terms: 0.1363636364 + x = -1.531952306 Solving 0.1363636364 + x = -1.531952306 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + x = -1.531952306 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + x = -1.531952306 + -0.1363636364 x = -1.531952306 + -0.1363636364 Combine like terms: -1.531952306 + -0.1363636364 = -1.6683159424 x = -1.6683159424 Simplifying x = -1.6683159424Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3955886696, -1.6683159424}
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